A polygon is any closed shape made of straight sides — triangles, rectangles, hexagons, and everything in between. Every polygon question boils down to two measurements: perimeter (the fence around it) and area (the carpet inside it). The perimeter rule is universal; the area formulas change by shape. Here’s the complete toolkit.
Perimeter vs. Area: Keep Them Straight
- Perimeter (linear units): the total distance around the boundary. Measured in cm, ft, m.
- Area (square units): the two-dimensional space enclosed inside. Measured in cm², ft², m².
If your polygon contains right-triangle segments with a missing side, review our guide on pythagorean theorem step-by-step first. For curved or continuous regions on coordinate grids, see our tutorial on definite integrals and area under curves. You can verify multi-step calculations with the automated CalcSolver tool.
The Universal Perimeter Rule
For any polygon, the perimeter is just the sum of all its outer sides:
Perimeter = Side₁ + Side₂ + Side₃ + … + Side_n
For a regular polygon (all sides equal), shortcut it: Perimeter = n × s, where n is the number of sides and s is one side’s length.
Area Formulas for Common Polygons
Triangles
- Perimeter: a + b + c
- Area: (base × height) ÷ 2
The height must be perpendicular to the base — a true 90° altitude, not the slanted side. Using the slant side is the classic way to overstate a triangle’s area.
Rectangles and parallelograms
- Rectangle perimeter: 2 × (length + width)
- Rectangle area: length × width
- Parallelogram area: base × perpendicular height
Trapezoids
A quadrilateral with one pair of parallel sides (base₁ and base₂):
- Perimeter: side₁ + side₂ + side₃ + side₄
- Area: ((base₁ + base₂) ÷ 2) × height — the average of the two bases times the height between them
Worked Example 1: Trapezoid
Find the perimeter and area of a trapezoid with bases b₁ = 14 cm and b₂ = 8 cm, non-parallel sides 5 cm and 6 cm, and perpendicular height 4 cm.
Step 1: Perimeter
P = 14 + 8 + 5 + 6 = 33 cm
Step 2: Area
A = ((14 + 8) ÷ 2) × 4 = (22 ÷ 2) × 4 = 11 × 4 = 44 cm²
Regular Polygons: The Apothem Formula
Regular pentagons, hexagons, and octagons have equal sides and equal angles. Their area uses the apothem — the perpendicular distance from the center to the midpoint of any side:
Area = (Perimeter × Apothem)/(2)
Worked Example 2: Regular hexagon
Find the area of a regular hexagon with side length 10 ft and apothem 8.66 ft.
- Perimeter: 6 × 10 = 60 ft
- Area: (60 × 8.66) ÷ 2 = 519.6 ÷ 2 = 259.8 ft²
Worked Example 3: Irregular Shape (Decomposition)
An L-shaped garden is a 9 m × 6 m rectangle with a 4 m × 3 m rectangular section cut out of one corner. Find its area and perimeter.
Area: subtract the missing piece
Full rectangle: 9 × 6 = 54 m². Cutout: 4 × 3 = 12 m². Area = 54 − 12 = 42 m².
Perimeter: count every exterior edge
The outer rectangle’s perimeter is 2 × (9 + 6) = 30 m, but the notch adds its two inner edges twice over: 2 × (4 + 3) = 14 m. Total perimeter = 30 + 14 = 44 m.
The lesson: for perimeter, walk the actual boundary — every inward jog adds length. For area, adding or subtracting whole rectangles is always the cleanest path.
3 Mistakes to Avoid
- Using slant height instead of perpendicular height: triangles, parallelograms, and trapezoids all demand the 90° altitude. The slanted side is always longer and always wrong here.
- Forgetting square units: perimeter is cm, area is cm². Dropping the ² on area is one of the most common exam deductions.
- Counting interior dividers in the perimeter: when you split a shape to find its area, those dividing lines are scaffolding — they are not part of the boundary.
When a polygon problem looks messy, decompose it. Almost every “impossible” shape is two or three friendly rectangles and triangles wearing a disguise.